Generalizations of the short pulse equation. (English) Zbl 1393.37076
The authors consider a wide class of second-order nonlinear partial differential equations which reduces to the short pulse equation for specific choices of the coefficients:
\[
u_{xt}=u+c_0 u^2+c_1 uu_x+ c_2 uu_{xx}+c_3u_x^2 +d_0u^3+d_1u^2u_x+d_2u^2u_{xx}+d_3uu_x^2.
\]
Based on previous results on the integrability of the short pulse equation, the authors analyze the integrability of the above general equation. They prove that if higher symmetries can be computed, then the above general equation gives rise to a list of integrable equations. For each listed equation, they give Hamiltonian structures, conservation laws and Lax representations.
Reviewer: Nilay Duruk Mutlubas (Tuzla)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35G20 | Nonlinear higher-order PDEs |
| 37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |
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